Understanding One-Way ANOVA & Assumptions - Analysis of Variance (AOV) - Prof. Brian C. De, Exams of Statistics

Download Understanding One-Way ANOVA & Assumptions - Analysis of Variance (AOV) - Prof. Brian C. De and more Exams Statistics in PDF only on Docsity! Introduction to Analysis of Variance (AOV, ANOVA) R. A. Fisher, 1920s 1-way AOV Idea: examine differences in means among several populations H : (common mean)! " # %. . .œ œ â œ H : (means are+ " # %. . .Á Á â Á different somewhere) Important assumption: all populations have a common variance ( )5# Examples 1. Experimental setting Potted plants in a greenhouse; 4 fertilizer mixes; randomly select 5 plants for each treatment ] œ 4 4 œ34 growth yield of the th plant ( 1, 2, 3, 4, 5) treated with mix ( 1, 2, 3, 4)3 3 œ normal , ] µ"4 " #a b. 5 normal , ] µ#4 # #a b. 5 normal , ] µ$4 $ #a b. 5 normal , ] µ%4 % #a b. 5 Under H , . Separate means area . . ." # >Á Á â Á needed to describe the data. Estimate them with , ,C C1 1"† #† ..., . Estimate with the pooled estimator (usingC1>† #5 separate means for each group: = œ 8 1 = 5 8 1 = 5â5 8 1 = 8 1 >[ # "" # > # # # # > X a b a b a b1 1 1 Numerator: SSW “within-sample sumDD 3 4 a bC 1 C œ134 3† # of squares” (sum of squared departures from their group means). Test statistic for hypothesis test based on following concept: does the model H meaningfully reduce thea amount of variability (noise) left over in the data? (i.e. are the extra mean parameters worthwhile?) Test statistic is essentially a comparison of with := =[# #X : H favored= = [ # X # small a : H favored= = ! [ # X # large Note: can show that TSS 1œ 8 1 = œ C 1 C1a b a bX 34X# †† #DD3 4 œ C 1 C 5 8 C 1 C1 1 1DD D 3 4 3 a b a b34 33† 3† ††# # SSW SSB (SSB: “between-sample sum of squares”) Also can show that = = œ œ Î 8 1 > 8 1 Î 8 1 8 1 > [ # X # X X X X SSW 1 SSW TSS 1 TSS a b a ba b a b œ 8 1 8 1 > 5 a ba bc dXX 1 SSW SSW SSB œ 8 1 8 1 > 5 Î a ba bc da bXX 1 1 SSB SSW œ 8 1 > 1 5 a b a b’ “ X 8 1> Î >1" >1 Î 8 1> 1 1 a b a ba b a bX X1 SSWSSB a b a b’ “ 8 1 > 1 5 0 X 8 1> >1 1 1 a ba bX 1 Here 0 œ œ Î > 1 " Î 8 1 > = = SSB SSW a ba bX F # [ # (between-sample variance) (within-sample variance) 0 : H favoredlarge a 0 : H favoredsmall ! If H is true, then! J œ µ > 1 8 1 > W W F # [ # XF 1, a b Hypothesis test Hypotheses: H : ! " # >. . . .œ œ â œ œ H : a . . ." # >Á Á â Á Test statistic: 0 œ œSSBSSW Î >1" Î 8 1> = = a ba bX F # [ # Rejection region: reject H if , the 100 1 th percentile of an! +0   0 1a b! F , distributiona b> 1 " 8 1 >X